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Ukrainian Mathematical Journal

, Volume 52, Issue 12, pp 1841–1857 | Cite as

Asymptotic Discontinuity of Smooth Solutions of Nonlinear q-Difference Equations

  • G. A. Derfel'
  • E. Yu. Romanenko
  • A. N. Sharkovsky
Article
  • 27 Downloads

Abstract

We investigate the asymptotic behavior of solutions of the simplest nonlinear q-difference equations having the form x(qt+ 1) = f(x(t)), q> 1, tR+. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), tR+. It is shown that, for “not very large” q> 1, the solutions of the q-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter qfor which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \)as T→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation.

Keywords

Asymptotic Behavior Difference Equation Asymptotic Property Smooth Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • G. A. Derfel'
    • 1
  • E. Yu. Romanenko
    • 2
  • A. N. Sharkovsky
    • 2
  1. 1.Ben-Gurion UniversityBeershebaIsrael
  2. 2.Institute of MathematicsUkrainian Academy of SciencesKiev

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